Applications of Homological Algebra to Toplogical Data Analysis

The use of large data sets in today’s world necessitates robust methods for classifying data. Topological data analysis (TDA) studies the shape of data collected in a geometric space. It has been applied to numerous data analysis problems, from glass manufacturing to cancer research.

TDA involves examining the data at different distances. TDA constructs a geometric space by connecting data points that are at most distance D away from each other. As D changes, topological features of the space appear and disappear. The collection of these features at each D is called a 1-module.

However, 1-modules can be susceptible to outliers. This can be addressed by introducing additional parameters, such as density, creating a 2-module. Representation theorists usually simplify 2-modules into an invariant that captures some features and allows comparison of different 2-modules. Invariants are created by approximating a 2-module by a sequence of simpler modules.

There are many different invariants, each with distinct capabilities and computational efficiencies. This project will develop efficient invariants and create a common framework for invariants. This will provide a method for selecting the appropriate invariant for a given application. We aim to develop new tools and frameworks, making TDA more powerful and widely applicable.

Faculty Supervisor:

Thomas Brüstle

Student:

Partner:

Kyoto University

Discipline:

Mathematics

Sector:

Education

University:

Université de Sherbrooke

Program: